Monotone coupling conjecture for random regular graphs

Let $n$ be the number of vertices, and let $0\le d_1\le d_2\le n-1$ be integers such that $d_1n$ and $d_2n$ are even. Write ${\mathcal G}(n,d_i)$ for the uniformly random $d_i$-regular graph. A coupling $(G_1,G_2)$ is a joint realization with these marginal distributions. Monotone coupling conjecture. Except for $(d_1,d_2)=(1,2)$ and $(d_1,d_2)=(n-3,n-2)$, there exists a coupling $(G_1,G_2)$ such that

$$G_1\sim {\mathcal G}(n,d_1),\qquad G_2\sim {\mathcal G}(n,d_2),$$

and

$$\operatorname{\mathbb P}(G_1\subseteq G_2)=1-o(1).$$

This conjecture asks for asymptotically almost-sure containment whenever the larger regular degree is at least the smaller one. The weakened sandwich results cited in the paper do not imply this assertion, even when the degree gap is large.